Does the theme of betrayal somehow influence the development of the character Winston?

Yes, betrayal influences Winston's development. He enters the Ministry of Love feeling assured that, whatever else happens, he will never betray Julia. While he is being tortured by O'Brien, he holds onto the idea that he has never once betrayed his beloved.
One night in prison, he dreams of being in the Golden Country. He wakes up with a start and cries out loud:

Julia! Julia! Julia, my love! Julia!

He senses her presence and knows she needs him:

In that moment he had loved her far more than he had ever done when they were together and free.

So far, he has remained loyal to her. But after this incident, O'Brien takes him to Room 101, where, afraid of being devoured by hungry rats, he betrays Julia by screaming to O'Brien to subject her to the rats:

Do it to Julia! Do it to Julia! Not me! Julia!

He saves his life with this betrayal, but something inside him breaks; he is never the same again:

Something was killed in your breast: burnt out, cauterized out.

When he and Julia happen to run into each other, they are no longer in love—she has also betrayed him. In fact, they dislike each other. Winston has changed: some part of him has died.